Slopes of 3-dimensional Subshifts of Finite Type

In this paper we study the directions of periodicity of three-dimensional subshifts of finite type (SFTs) and in particular their slopes. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope being the angle of the periodicity vectors. In this paper, we prove that any ${\mathrm{\Sigma }}_2^0$ set may be realized as a a set of slopes of an SFT.